#8837063

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#55218	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,10,9,5)(4,12,13,8)(11,15,14,16)(17,18)(19,21)(20,25)(22,30)(23,24)(26,31)(27,32)(28,29)(33,34)(35,36)(37,45)(38,39)(40,41)(42,46)(43,47)(44,48);(2,7)(4,13)(5,10)(11,14)(17,19,23,20)(18,21,24,25)(22,28,31,27)(26,32,30,29)(34,39)(35,41)(42,45)(43,48);(2,7)(3,9)(8,12)(11,14)(18,24)(19,20)(26,30)(27,28)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,3,6,9)(2,5,7,10)(4,14,13,11)(8,16,12,15)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,4)(2,8)(3,11)(5,15)(6,13)(7,12)(9,14)(10,16)(17,21,23,25)(18,20,24,19)(22,29,31,32)(26,28,30,27)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,5)(2,9)(3,7)(4,15)(6,10)(8,14)(11,12)(13,16)(17,22,23,31)(18,26,24,30)(19,27,20,28)(21,29,25,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8744259);